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Find X from cash flow diagram, 1% per month

Short answer

Convert 1% per month to effective annual rate 12.6825%. Equate present worth of outflows and inflows: 15,000 = X(P/F,12.68%,2) + X(P/F,12.68%,6) + 2X(P/F,12.68%,10). Result: X = Rs 7,970.

Find the value of X from the following figure. i = 1% per month. [Figure: Rs 15000 outflow at time 0; X received at 2, X received at 6, and 2X received at 10 years, as marked on the diagram]

Assumptions

The figure shows the times (2, 6, 10) in years, while the rate is 1% per month, so the equivalent yearly rate is needed: i = (1.01)^12 − 1 = 12.6825% per year. (If the times are actually months, use 1% directly; see the end.)

Cash flow diagram (text)

        X          X                2X
        ^          ^                ^
 -------+----------+----------------+--> time
 0      2          6                10
 |
 v
 15,000

Equation (present worth at time 0)

Outflow = Inflows: 15,000 = X (P/F, i, 2) + X (P/F, i, 6) + 2X (P/F, i, 10)

With i = 12.6825%:

  • (P/F, i, 2) = 1/(1.126825)² = 0.78757
  • (P/F, i, 6) = 1/(1.126825)⁶ = 0.48850
  • (P/F, i, 10) = 1/(1.126825)¹⁰ = 0.30299

15,000 = X (0.78757 + 0.48850 + 2 × 0.30299) = X (1.88206)

X = 15,000 / 1.88206 = Rs 7,970

Answer: X = about Rs 7,970.

If the points 2, 6, 10 are months at 1% per month, the factors are 0.98030, 0.94205 and 0.90529, so X = 15,000 / (0.98030 + 0.94205 + 2 × 0.90529) = 15,000 / 3.7329 = Rs 4,018.

✓ Answer checked by Claude Opus 5.5 (high effort)

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